Can't resist putting my 2¢ in:
Irene Goodnight, around 1908, possible earlier; words are kind of depressing, but the tune is lovely:
1910, Carrie Jacobs-Bond:
Precious Memories, 1925:
(This is hard; most of that time period was either ragtime (no words) or early gospel- "Will the Circle be Unbroken," 1907, "I'll Fly Away")
There might be some Louis Armstrong stuff for the 30s..."Dream a Little Dream?" Wiki says 1931
either skip the bridges, or invite the kids to play air trumpet?
"Wise Men Say," 1961, maybe a bit late for your gap, covered by everyone from Joan Baez to UB40:
Katie Melua's Sailboat Song, because it lends itself to inventing one's own lyrics in the manner of "Every Breath You Take" (every lawn you rake, every stripy snake, etc.) (2007):
And loads of Irish songs (REAL ones), but it's difficult to be sure of the dates and they're probably pre-1900. Let me know if you're interested. Need one from 17th century (possibly earlier)? It's very pretty and turned up again in 2008 or so:
(don't let the uilleann pipes put you off, it works great a capella.)
Regina Spektor? (2008 or so):
"All the Roadrunning" (mark knopfler with emmylou harris):
or Border Reiver:
My dad used to sing
"Home, home on the range,
where the beer and the cantaloupes play.
Where seldom is heard
a disparaging word
and the skies are not crowded all day."
(The range, of course, being the thing we cooked dinner on, in the event that I asked.)
"When the moon hits your eye
like a big pizza pie
I asked him what 'amore' was. He said "it's like an eel, only bigger." He loved to mess with my head :)